Statistics a Gentle Introduction Ch_3

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80

CHAPTER 3

STATISTICAL PARAMETERS

Measures of Central Tendency and Variation

• Understand and compute measures of central tendency

• Understand and compute measures of variation

• Learn the differences between klinkers and outliers and how to deal with them

• Learn how Tchebysheff’s theorem led to the development of the empirical rule

• Learn about the sampling distribution of means, the central limit theorem, and the standard errorof the mean

• Apply the empirical rule for making predictions

Chapter 3 Goals

MEASURES OF CENTRAL TENDENCY

In addition to graphs and tables of numbers, statisticians often use common parameters to

describe sets of numbers. There are two major categories of these parameters. One group ofparameters measures how a set of numbers is centered around a particular point on a line scale

or, in other words, where (around what value) the numbers bunch together. This category of

parameters is called measures of central tendency . You already know and have used the most

famous statistical parameter from this category, which is the mean or average.



Chapter 3. Statistical Parameters 81

The Mean

The mean is the arithmetic average of a set of scores. There are actually different kinds of means,

such as the harmonic mean (which will be discussed later in the book) and the geometric mean.

We will first deal with the arithmetic mean. The mean gives someone an idea where the centerlies for a set of scores. The arithmetic mean is obtained by taking the sum of all the numbers in

the set and dividing by the total number of scores in the set.

You already learned how to derive the mean in Chapter 1. Here again is the official formula

in proper statistical notation:

x

x

N

ii

k

=

∑=1

where

x = the mean,

xii

k

=

∑1

= the sum of all the scores in the set,

N = the number of scores or observations in the set.

Note that from this point on in the text, we will use the shorthand symbol ∑ x instead of the

proper xi

i

k

=

∑1

.

The mean has many important properties that make it useful. Probably its most attractive

quality is that it has a clear conceptual meaning. People almost automatically understand and

easily form a picture of an unseen set of numbers when the mean of that set is presented alone.

Another attractive quality is that the mathematical formula is simple and easy. It involves only

adding, counting, and dividing. The mean also has some more complicated mathematical prop-

erties that also make it highly useful in more advanced statistical settings such as inferential

statistics. One of these properties is that the mean of a sample is said to be an unbiased estima-

tor of the population mean. But first, let us back up a bit.

The branch of statistics known as inferential statistics involves making inferences or

guesses from a sample about a population. As members of society, we continually make deci-

sions, some big (such as what school to attend, who to marry, or whether to have an opera-

tion) and some little (such as what clothes to wear or what brand of soda to buy). We hope

that our big decisions are based on sound research. For example, if we decide to take a drug

to lower high blood pressure, we hope that the mean response of the participants to the drug

is not just true of the sample but also of the population, that is, all people who could takethe drug for high blood pressure. So if we take the mean blood pressure of a sample of

patients after taking the drug, we hope that it will serve as an unbiased estimator of the

population mean; that is, the mean of a sample should have no tendency to overestimate or

underestimate the population mean, µ (which is also written mu and pronounced mew).



82 STATISTICS: A GENTLE INTRODUCTION

Thus, if consecutive random samples are drawn from a larger population of numbers, each

sample mean is just as likely to be above µ as it is to be below µ. This property is also useful

because it means that the population formula for µ is the same as the sample formula for x .

These formulas are as follows:

Sample Population

Meanx

x

N = ∑

µ = ∑ x

N

The Median

Please think back to when you heard income or wealth reports in the United States. Can you

recall if they reported the mean or the median income (or wealth)? More than likely, you heard

reports about the median and not the mean. Although the mean is the mostly widely used

measure of central tendency, it is not always appropriate to use it. There may be many situations

where the median may be a better measure of central tendency. The median value in a set of

numbers is that value that divides the set into equal halves when all the numbers have been

ordered from lowest to highest. Thus, when the median value has been derived, half of all the

numbers in the set should be above that score, and half should be below that score. The reason

that the median is used in reports about income or wealth in the United States is that income

and wealth are unevenly distributed (i.e., they are not normally distributed). A plot of the fre-

quency distribution of income or wealth would reveal a skewed distribution. Now, would the

resulting distribution be positively skewed or negatively skewed? It is easier to answer that

question if you think in terms of wealth. Are most people in the United States wealthy and

there’s just a few outlying poor people, or do most people have modest wealth and a few

people own a huge amount? Of course, the latter is true. It is often claimed that the wealthiest

3% of U.S. citizens own about 40% of the total wealth. Thus, wealth is positively skewed. A mean

value of wealth would be skewed or drawn to the right by a few higher values, giving the

appearance that the average person in the United States is far wealthier than he or she really is.

In all skewed distributions, the mean is strongly influenced by the few high or low scores. Thus,

the mean may not truthfully represent the central tendency of the set of scores because it has

been raised (or lowered) by a few outlying scores. In these cases, the median would be a bettermeasure of central tendency. The formula for obtaining the median for a set of scores will vary

depending on the nature of the ordered set of scores. The following two methods can be used

in many situations.




Method 1

When the scores are ordered from lowest to highest and there are an odd number of scores, the

middle value will be the median score. For example, examine the following set of scores:

7, 9, 12, 13, 17, 20, 22

Because there are seven scores and 7 is an odd number, then the middle score will be the

median value. Thus, 13 is the median score. To check whether this is true, look to see whether

there is exactly the same number of scores above and below 13. In this case, there are three

scores above 13 (17, 20, and 22) and three scores below 13 (7, 9, and 12).

Method 2

When the scores are ordered from lowest to highest and there are an even number of scores,

the point midway between the two middle values will be the median score. For example, exam-ine the following set of scores:

2, 3, 5, 6, 8, 10

There are six scores, and 6 is an even number; therefore, take the average of 5 and 6, which

is 5.5, and that will be the median value. Notice that in this case, the median value is a hypo-

thetical number that is not in the set of numbers. Let us change the previous set of numbers

slightly and find the median:

2, 3, 5, 7, 8, 10

In this case, 5 and 7 are the two middle values, and their average is 6; therefore, the median

in this set of scores is 6. Let us obtain the mean for this last set, and that is 5.8. In this set, the

median is actually slightly higher than the mean. Overall, however, there is not much of a differ-

ence between these two measures of central tendency. The reason for this is that the numbers

are relatively evenly distributed in the set. If the population from which this sample was drawn

is normally distributed (and not skewed), then the mean and the median of the sample will be

about the same value. In a perfectly normally distributed sample, the mean and the median will

be exactly the same value.

Now, let us change the last set of numbers once again:

2, 3, 5, 7, 8, 29

Now the mean for this set of numbers is 9.0, and the median remains 6. Notice that the

mean value was skewed toward the single highest value (29), while the median value was not




affected at all by the skewed value. The mean in this case is not a good measure of central ten-

dency because five of the six numbers in the set fall below the mean of 9.0. Thus, the median

may be a better measure of central tendency if the set of numbers has a skewed distribution. See

Figure 3.1 for graphic examples.

Figure 3.1 Graphic Examples of Distributions Showing Median and Mean

F

r e q u e n c y

Mean Median

F

r e q u e n c y

MeanMedian

If there are ties at the median value when you use either of the two previous methods, then

you should consult an advanced statistics text for a third median formula, which is much more

complicated than the previous two methods. For example, examine the following set:

2, 3, 5, 5, 5, 10

There are an even number of scores in this set, and normally we would take the average of

the two middle values. However, there are three 5s, and that constitutes a tie at the median

value. Notice that if we used 5 as the value of the median, there is one score above the value 5,

and there are two scores below 5. Therefore, 5 is not the correct median value. It is actually 4.54,

which is confusing (because there are two scores below that value and four above it); however,

it is the correct theoretical median.

The Mode

The mode is a third measure of central tendency. The mode score is the most frequently occur-

ring number in a set of scores. In the previous set of numbers, 5 would be the mode score

because it occurs at a greater frequency than any other number in that set. Notice that the mode




score in that set is 5, and the frequency (how many are there) of the mode score is 3 because

there are three 5s in the set.

It is also possible to have two or more mode scores in a set of numbers. For example, exam-

ine this set:

2, 3, 3, 3, 4, 5, 6, 6, 6, 8

In this set, there are two modes: One mode score is 3, and the other mode score is 6. The

frequency of both mode scores is 3. A distribution that has two different modes is said to be

bimodally distributed. The mode score can change drastically across different samples, and thus

it is not a particularly good overall measure of central tendency. The mode probably has its great-

est value as a measure with nominal or categorical scales. For example, we might report in a

study that there were 18 male and 14 female participants. Although it may appear obvious and

highly intuitive, we know that there were more male than female participants because we can

see that 18 (males) is the mode score.

CHOOSING BETWEEN MEASURESOF CENTRAL TENDENCY

In one of my older journal articles, “Dreams of the Dying” (Coolidge & Fish, 1983), an under-

graduate student and I obtained dream reports from 14 dying cancer patients. In the method

section of the article, we reported the subjects’ ages. Rather than list the ages for all 14 subjects,

what category of statistical parameters would be appropriate? Of course, it would be the cate-

gory of measures of central tendency. The following numbers represent the subjects’ ages at the

time of their deaths:

28, 34, 40, 40, 42, 43, 45, 48, 59, 59, 63, 63, 81, 88

The mean for this set of scores is 52.4, and the median is 46.5 [(45 + 48)/2]. Typically, a

researcher would not report both the mean and the median, so which of the two measures

would be reported? A graph of the frequency distribution (by intervals of 10 years) shows that

the distribution appears to be skewed right. See Figure 3.2 for two versions of the frequency

distribution.

Because of this obvious skew, the mean is being pulled by the extreme scores of 81 and 88.

Thus, in this case, we reported the median age of the subjects instead of the mean.

In most statistical situations, the mean is the most commonly used measure of central ten-

dency. Besides its ability to be algebraically manipulated, which allows it to be used in conjunc-

tion with other statistical formulas and procedures, the mean also is resistant to variations across

different samples.




KLINKERS AND OUTLIERS

Sometimes when we are gathering data, we may have

equipment failure, or in a consumer preference study, we

may have a subject who does not speak English. A datum

from these situations may be simply wrong or clearly inap-

propriate. Abelson (1995) labels these numbers klinkers.

Abelson argues that when it is clearly inappropriate to

keep klinkers, they should be thrown out, and additional

data should be gathered in their place. Can I do this? Is this

ethical? You may legitimately ask yourself these questions.

Tukey (1969) and Abelson both warn against becoming too

“stuffy” and against the “sanctification” of statistical rules.

If equipment failure has clearly led to an aberrant score, or

if some participants in a survey failed to answer some ques-

tions because they did not speak English, then our course

is clear. Keeping these data would ruin the true spirit of the

statistical investigative process. We are not being statisti-

cally conservative but foolish if we kept such data.

The other type of aberrant score in our data is calledan outlier . Outliers are deviant scores that have been

legitimately gathered and are not due to equipment fail-

ures. While we may legitimately throw out klinkers, outliers are a much murkier issue. For

example, the sports section of a newspaper reported that the Chicago Bulls were the highest

Figure 3.2 Two Versions of Frequency Distribution

2 0 – 2 9

3 0 – 3 9

4 0 – 4 9

5 0 – 5 9

6 0 – 6 9

7 0 – 7 9

8 0 – 8 9

Age Intervals

F r e q u e n c y

1

2

3

4

5

6

2 0 – 3

4

3 5 – 4 9

5 0 – 6

4

6 5 – 7 9

8 0 – 9

4

Age Intervals

F r e q u e n c y

1

2

3

4

5

6

Michael Jordan (born February17, 1963)

Source: Photo © Steve Lipofsky, www.Basketballphoto.com. Used with thekind permission of the photographer.




paid team (1996–1997) in the National Basketball Association, with an average salary of $4.48

million. However, let’s examine the salaries and see if the mean is an accurate measure of the

13 Bulls players (Table 3.1).

As a measure of central tendency, the mean Bulls’s salary is misleading. Only two players

have salaries above the mean, Michael Jordan and Dennis Rodman, while 11 players are belowthe mean. A better measure of central tendency for these data would be the median. Randy

Brown’s salary of $1,300,000 would be the median salary since 6 players have salaries above that

number and 6 players are below that number. It is clear that Michael Jordan’s salary is skewing

the mean, and his datum would probably be considered an outlier. Without Jordan’s salary, the

Bulls’s average pay would be about $2,300,000, which is less than half of what the Bulls’s average

salary would be with his salary. Furthermore, to show how his salary skews the mean, Michael

Jordan made about $2,000,000 more than the whole rest of the team’s salaries combined!

We will revisit the issue of outliers later in this chapter. Notice in Jordan’s case, we did not

eliminate his datum as an outlier. First, we identified it as an outlier (because of its effect of skew-

ing the mean of the distribution) and reported the median salary instead. Second, we analyzed

the data (team salaries) both ways, with Jordan’s salary and without, and we reported both sta-tistical analyses. There is no pat answer to dealing with outliers. However, the latter approach

(analyzing the data both ways and reporting both of the analyses) may be considered semicon-

servative. The more conservative position would be to analyze the data and report them with

the outlier and have no other alternative analyses without the outlier.

Player Salary ($)

Jordan 30,140,000Rodman 9,000,000

Kukoc 3,960,000

Harper 3,840,000

Longley 2,790,000

Pippen 2,250,000

Brown 1,300,000

Simpkins 1,040,000

Parish 1,000,000

Wennington 1,000,000Kerr 750,000

Caffey 700,000

Buechler 500,000

Table 3.1 Salary Distribution With Outlier




UNCERTAIN OR EQUIVOCAL RESULTS

One of my colleagues sums up statistics to a single phrase, “it’s about measurement.” Often, as

in the case of klinkers and outliers, measurement is not as straightforward as it would appear.

Sometimes measurement issues arise because there is a problem with the measurement proce-

dures or measurement. When the results of the measurement are unclear, uncertain, or unread-

able, they are said to be equivocal.

Intermediate results occur when a test outcome is neither positive nor negative but falls

between these two conditions. An example might be a home pregnancy kit that turns red if a

woman is pregnant and stays white if she is not. If it turns pink, how are the results to be inter-

preted? Indeterminate results may occur when a test outcome is neither positive nor negative

and does not fall between these two conditions. Perhaps, in the pregnancy kit example, this

might occur if the kit turns green. Finally, it may be said that uninterpretable results have

occurred when a test has not been given according to its correct instructions or directions, or

when it yields values that are completely out of range.

MEASURES OF VARIATION

The second major category of statistical parameters is measures of variation. Measures of varia-

tion tell us how far the numbers are scattered about the center value of the set. They are also called

measures of dispersion. There are three common parameters of variation: the range, standard

deviation, and variance. While measures of central tendency are indispensable in statistics, mea-

sures of variation provide another important yet different picture of a distribution of numbers. For

example, have you heard of the warning against trying to swim across a lake that averages only 3

feet deep? While the mean does give a picture that the lake on the whole is shallow, we intuitivelyknow that there is danger because while the lake may average 3 feet in depth, there may be much

deeper places as well as much shallower places. Thus, measures of central tendency are useful in

understanding how scores cluster about a center value, and measures of variation are useful in

understanding how far, wide, or deep the high scores are scattered about the center value.

The Range

The range is the simplest of the measures of variation. The range describes the difference

between the lowest score and the highest score in a set of numbers. Typically, statisticians do not

actually report the range value, but they do state the lowest and highest scores. For example,given the set of scores 85, 90, 92, 98, 100, 110, 122, the range value is 122 − 85 = 37. Therefore,

the range value for this set of scores is 37, the lowest score is 85, and the highest score is 122.

It is also important to note that the mean for this set is 99.6, and the median score is 98.

However, neither of these measures of central tendency tells us how far the numbers are from




this center value. The set of numbers 95, 96, 97, 98, 99, 103, 109 would also have a mean of 99.6

and a median of 98. However, notice how the range, as a measure of variation, tells us that in

the first set of numbers, they are widely distributed about the center (range = 37), while in the

second set, the range is only 14 (109 − 95 = 14). Thus, the second set varies less about its cen-

ter than the first set.Let us refer back to the ages of the dream subjects in the previously mentioned study.

Although the mean of the subjects’ ages was 52.4, we have no idea how the ages are distributed

about the mean. In fact, because the median was reported, the reader may even suspect that the

ages are not evenly distributed about the center. The reader might correctly guess that the ages

might be skewed, but the reader would not know whether the ages were positively or negatively

skewed. In this case, the range might be useful. For example, it might be reported that the scores

ranged from a low of 28 to a high of 88 (although the range value itself, which is 60, might be of

little conceptual use). Although the range is useful, the other two measures of variation, stan-

dard deviation and variance, are used far more frequently. The range is useful as a preliminary

descriptive statistic, but it is not useful in more complicated statistical procedures, and it varies

too much as a function of sample size (the range goes up when the sample size goes up). Therange also depends only on the highest and lowest scores (all of the other scores in the set do

not matter), and a single aberrant score can affect the range dramatically.

The Standard Deviation

The standard deviation is a veritable bulwark in the sea of statistics. Along with the mean, the

standard deviation is a theoretical cornerstone in inferential statistics. The standard deviation

gives an approximate picture of the average amount each number in a set varies from the center

value. To appreciate the standard deviation, let us work with the idea of the average deviation.

Let us work with a small subsample of the ages of the dream subjects:

28, 42, 48, 59, 63

Their mean is 48.0. Let us see how far each number is from the mean.

Each Number ( xi) Mean ( x ) Distance From Mean ( x xi − )

28 48 −20

42 48 −6

48 48 0

59 48 +11

63 48 +15




Note that the positive and negative signs tell us whether an individual number is above or below

the mean. The size or magnitude of the distance score tells us how far that number is from the mean.

To get the average deviation for this set of scores, we would normally sum the five distance

values and divide by 5 because there were five scores. In this case, however, if we sum −20, −6,

0, +11, and +15, we would get 0 (zero), and 0 divided by 5 is 0.One solution to this dilemma would be to take the absolute value of each distance. This

means that we would ignore the negative signs. If we now try to average the absolute values of

the distances, we would obtain the sum of 20, 6, 0, 11, and 15, which is 52, and 52 divided by 5

is 10.4. Now, we have a picture of the average amount each number varies from the mean, and

that number is 10.4.

The average of the absolute values of the deviations has been used as a measure of variation,

but statisticians have preferred the standard deviation as a better measure of variation, particu-

larly in inferential statistics. One reason for this preference, especially among mathematicians, is

that the absolute value formula cannot be manipulated algebraically.

The formula for the standard deviation for a population is

σ = −∑( ) x x

N

i

2

Note that σ or sigma represents the population value of the standard deviation. You previ-

ously learned Σ as the command to sum numbers together. Σ is the capital Greek letter, and σ is

the lowercase Greek letter. Also note that although they are pronounced the same, they have

radically different meanings. The sign Σ is actually in the imperative mode; that is, it states a

command (to sum a group of numbers). The other value σ is in the declarative mode; that is, it

states a fact (it represents the population value of the standard deviation).

The sample standard deviation has been shown to be a biased estimator of the population value, and consequently, there is bad news and good news. The bad news is that there are two dif-

ferent formulas, one for the sample standard deviation and one for the population standard devia-

tion. The good news is that statisticians do not often work with a population of numbers. They

typically only work with samples and make inferences about the populations from which they were

drawn. Therefore, we will only use the sample formula. The two formulas are presented as follows:

Sample Population

Standard deviation

S x x

N

i = −∑

−

( )2

1

σ = −∑ ( )x x

N

i

2

where S (capital English letter S) stands for the sample standard deviation.




CORRECTING FOR BIAS IN THESAMPLE STANDARD DEVIATION

Notice that the two formulas only differ in their denominators. The sample formula has N − 1 in

the denominator, and the population has only N. When it was determined that the original for-

mula (containing only N in the denominator) consistently underestimated the population value

when applied to samples, the correction −1 was added to correct for the bias. Note that the

correction makes the numerator larger (because the correction makes the denominator

smaller), and that makes the value of the sample standard deviation larger (if we divide the

numerator by a large number, then it makes the numerator smaller; if we divide the numerator

by a smaller number, then that makes the numerator larger). The correction for bias has its great-

est effect in smaller samples, for example, dividing by 9 instead of 10. In larger samples, the

power of the correction is diminished, yet statisticians still leave the correction in the formula,

even in the largest samples.

HOW THE SQUARE ROOT OF x2 IS ALMOST EQUIVALENT TOTAKING THE ABSOLUTE VALUE OF x

As mentioned earlier, statisticians first used the absolute value method of obtaining the average

deviation as a measure of variation. However, squaring a number and then taking the square root

of that number also removes negative signs while maintaining the value of the distance from the

mean for that number. For example, if we have a set of numbers with a mean of 4 and our lowest

number in the set is 2, then 2 − 4 = −2. If we square −2, we get 4, and the square root of 4 = 2.

Therefore, we have removed the negative sign and preserved the original value of the distance

from the mean. Thus, when we observe the standard deviation formula, we see that the numer-

ator is squared, and we take the square root of the final value. However, if we take a set of num-

bers, the absolute value method for obtaining the standard deviation and the square root of the

squares method will yield similar but not identical results. The square root of the square method

has the mathematical property of weighting numbers that are farther from the mean more heav-

ily. Thus, given the previous subset of numbers

28, 42, 48, 59, 63

the absolute value method for standard deviation (without the correction for bias) yielded a

value of 10.4, while the value of the square root of the squares method is 12.5.




THE COMPUTATIONAL FORMULAFOR STANDARD DEVIATION

One other refinement of the standard deviation formula has also been made, and this change

makes the standard deviation easier to compute. The following is the computational formula for

the sample standard deviation:

S

x x

N

N =

∑ − ∑

−

22

1

( )

Remember that this computational formula is exactly equal to the theoretical formula pre-

sented earlier. The computational formula is simply easier to compute. The theoretical formula

requires going through the entire data three times: once to obtain the mean, once again to

subtract the mean from each number in the set, and a third time to square and add the numbers

together. Note that on most calculators, ∑ x and ∑ x2 can be performed at the same time; thus,

the set of numbers will only have to be entered in once. Of course, many calculators can obtain

the sample standard deviation or the population value with just a single button (after entering

all of the data). You may wish to practice your algebra, nonetheless, with the computational

formula and check your final answer with the automatic buttons on your calculator afterward.

Later in the course, you will be required to pool standard deviations, and the automatic standard

deviation buttons of your calculator will not be of use. Your algebraic skills will be required, so

it would be good to practice them now.

THE VARIANCEThe variance is a third measure of variation. It has an intimate mathematical relationship with stan-

dard deviation. Variance is defined as the average of the square of the deviations of a set of scores

from their mean. In other words, we use the same formula as we did for the standard deviation,

except that we do not take the square root of the final value. The formulas are presented as follows:

Sample Population

VarianceS

x x

N

i 22

1=

−∑

−

( )σ

22

= −∑ ( )x x

N

i

Statisticians frequently talk about the variance of a set of data, and it is an often-used

parameter in inferential statistics. However, it has some conceptual drawbacks. One of them is




that the formula for variance leaves the units of measurement squared. For example, if we said

that the standard deviation for shoe sizes is 2 inches, it would have a clear conceptual meaning.

However, imagine if we said the variance for shoe sizes is 4 inches squared. What in the world

does “inches squared” mean for a shoe size? This conceptual drawback is one of the reasons

that the concept of standard deviation is more popular in descriptive and inferential statistics.

THE SAMPLING DISTRIBUTION OF MEANS,THE CENTRAL LIMIT THEOREM, AND THE STANDARD ERROR OF THE MEAN

A sampling distribution is a theoretical frequency distribution that is based on repeated

(a large number of times) sampling of n-sized samples from a given population. If the means for

each of these many samples are formed into a frequency distribution, the result is a sampling

distribution. In 1810, French mathematician Pierre LaPlace (1749–1827) formulated the centrallimit theorem, which postulated that if a population is normally distributed, the sampling dis-

tribution of the means will also be normally distributed. However, more important, even if the

scores in the population are not normally distributed, the sampling distribution of means will still

be normally distributed. The latter idea is an important concept, especially if you take additional

statistics classes. The standard deviation of a sampling distribution of means is called the stan-

dard error of the mean. As the sample size of the repeated samples becomes larger, the stan-

dard error of the mean becomes smaller. Also, as the sample size increases, the sampling

distribution approaches a normal distribution. At what size a sample does the sampling distribu-

tion approach a normal distribution? Most statisticians have decided upon n = 30. It is important

to note, however, if the population from which the sample is drawn is normal, then any size

sample will lead to a normally distributed sampling distribution. The characteristics of the sam-pling distribution serve as an important foundation for hypothesis testing and the statistical tests

presented later in this book. However, at this point in a statistics course, it might be important to

remember at least this: In inferential statistics, a sample size becomes arbitrarily large at n ≥ 30.

THE USE OF THE STANDARD DEVIATION FOR PREDICTION

Pafrutti Tchebysheff (1821–1894), a Russian mathematician, developed a theorem, which ulti-

mately led to many practical applications of the standard deviation. Tchebysheff ’s theorem could

be applied to samples or populations, and it stated that specific predictions could be made about

how many numbers in a set would fall within a standard deviation or standard deviations fromthe mean. However, the theorem was found to be conservative, and statisticians developed the

notion of the empirical rule. The empirical rule holds only for normal distribution or relatively

mound-shaped distributions.




The empirical rule predicts the following:

1. Approximately 68% of all numbers in a set will fall within ±1 standard deviation of the mean.

2. Approximately 95% of all numbers in a set will fall within ±2 standard deviations of the mean.

3. Approximately 99% of all numbers in a set will fall within ±3 standard deviations of the mean.

For example, let us return to the ages of the subjects in the dream study previously mentioned:

28, 34, 40, 40, 42, 43, 45, 48, 59, 59, 63, 63, 81, 88

The mean is 52.4. The sample standard deviation computational formula is as follows:

S

x x

N

N = ∑ −

∑

−= − = −

= −

22 2

142287

733

1413

42287

537289

1413

42287 38

( ) ( )

3377 7857

13

3909 2143

13300 7088 17 34

. .. .= = =

Thus, S = 17.34.

Now, let us see what predictions the empirical rule will make regarding this mean and stan-

dard deviation.

1. x S + = + =1 52 4 17 3 69 7. . .

x S − = − =1 52 4 17 3 35 1. . .

Thus, the empirical rule predicts that approximately 68% of all the numbers will fall within

this range of 35.1 years old to 69.7 years old.

If we examine the data, we find that 10 of the 14 numbers are within that range, and 10/14

is about 70%. We find, therefore, that the empirical rule was relatively accurate for this distribu-

tion of numbers.

2. x S + = +

( ) = + =2 52 4 2 17 3 52 4 34 6 87 0. . . . .

x S – . – . . – . .2 52 4 2 17 3 52 4= ( ) = =34 6 17 8




Inspection of the data reveals that 13 of the 14 numbers in the set fall within two standard

deviations of the mean, or approximately 93%. The empirical rule predicted about 95%; thus, it

was again relatively accurate for these data.

3. x S + = + ( ) = + =3 52 4 3 17 3 52 4 51 9 104 3. . . . .

x S – . – . . – . .3 52 4 3 17 3 52 4 51 9 5= ( ) = = 0

All 14 of the 14 total numbers fall within three standard deviations of the mean. The empir-

ical rule predicted 99%, and again we see the rule was relatively accurate, which implies that this

group of numbers was relatively normally distributed, as it was close to the predictions of the

empirical rule at ±1, ±2, and ±3 standard deviations of the mean.

PRACTICAL USES OF THE EMPIRICAL

RULE: AS A DEFINITION OF AN OUTLIER

Some statisticians use the empirical rule to define outliers. For example, a few statisticians define

an outlier as a score in the set that falls outside of ±3 standard deviations of the mean. However,

caution is still urged before deciding to eliminate even these scores from an analysis because

although they may be improbable, they still occur.

PRACTICAL USES OF THE EMPIRICAL RULE:PREDICTION AND IQ TESTS

The empirical rule has great practical significance in the social sciences and other areas. For

example, IQ scores (on Wechsler’s IQ tests) have a theoretical mean of 100 and a standard

deviation of 15. Therefore, we can predict with a reasonable degree of accuracy that 68% of a

random sample of normal people taking the test should have IQs between 85 and 115.

Furthermore, only 5% of this sample should have an IQ below 70 or above 130 because the

empirical rule predicted that 95% would fall within two standard deviations of the mean.

Because IQ scores are assumed to be normally distributed and both tails of the distribution are

symmetrical, we can predict that 2.5% of people will have an IQ less than 70 and 2.5% will have

IQs greater than 130.

What percentage of people will have IQs greater than 145? An IQ of 145 is exactly three

standard deviations above the mean. The empirical rule predicts that 99% should fall within ±3standard deviations of the mean. Therefore, of the 1.0% who fall above 145 or below 55, 0.5%

will have IQs above 145.




SOME FURTHER COMMENTS

The two categories of parameters, measures of central tendency and measures of variation, are

important in both simple descriptive statistics and in inferential statistics. As presented, you

have seen how parameters from both categories are necessary to describe data. Remember

that the purpose of statistics is to summarize numbers clearly and concisely. The parameters,

mean and standard deviation, frequently accomplish these two goals, and a parameter from

each category is necessary to describe data. However, being able to understand the data

clearly is the most important goal of statistics. Thus, not always will the mean and standard

deviation be the appropriate parameters to describe data. Sometimes, the median will make

better sense of the data, and most measures of variability do not make sense for nominal or

categorical data.

HISTORY TRIVIA

Fisher to Eels

Ronald A. Fisher (1890–1962) received an undergraduate degree in astronomy in England. After

graduation, he worked as a statistician and taught mathematics. At the age of 29, he was hired at

an agricultural experimental station. Part of the lure of the position was that they had gathered

approximately 70 years of data on wheat crop yields and weather conditions. The director of the

station wanted to see if Fisher could statistically analyze the data and make some conclusions.

Fisher kept the position for 14 years. Consequently, modern statistics came to develop some

strong theoretical “roots” in the science of agriculture.

Fisher wrote two classic books on statistics, published in 1925 ( Statistical Methods for

Research Workers ) and 1935 ( The Design of Experiments ). He also gave modern statistics two

of its three most frequently used statistical tests, t tests and analysis of variance. Later in his

career, in 1954, he published an interesting story of a scientific discovery about eels and the

standard deviation. The story is as follows:

Johannes Schmidt was an ichthyologist (one who studies fish) and biometrician (one who

applies mathematical and statistical theory to biology). One of his topics of interest was the

number of vertebrae in various species of fish. By establishing means and standard deviations for

the number of vertebrae, he was able to differentiate between samples of the same species

depending on where they were spawned. In some cases, he could even differentiate between

two samples from different parts of a fjord or bay.

However, with eels, he found approximately the same mean and same large standard devia-tion from samples from all over Europe, Iceland, and Egypt. Therefore, he inferred that eels from

all these different places had the same breeding ground in the ocean. A research expedition in

the Western Atlantic Ocean subsequently confirmed his speculation. In fact, Fisher notes, the

expedition found a different species of eel larvae for eels of the eastern rivers of North America

and the Gulf of Mexico.




Key Terms, Symbols, and Definitions

Central limit theorem —A mathematicalproposition that states that if a population’sscores are normally distributed, the samplingdistribution of means will also be normally dis-tributed, and if the scores in the population arenot normally distributed, the sampling distribu-tion of means will still be normally distributed.

Empirical rule —A rule of thumb that dic-tates that approximately 68% of all scores willfall within plus or minus one standard devia-tion of the mean in a normal distribution, 95% will fall within plus or minus two standarddeviations, and 99% will fall within plus or

minus three standard deviations of the mean.Equivocal results —Test results that areunclear, uncertain, or unreadable.

Indeterminate results —A condition when atest’s outcome is neither positive nor negativeand does not fall between these two conditions.

Intermediate results —A condition when atest’s outcome is neither positive nor negativebut falls between these two conditions.

Klinkers —A term used for a datum that isclearly wrong or inappropriate (e.g., a num-

ber not on the measuring scale).

Mean ( x –

) —The arithmetical average of agroup of scores.

Measures of central tendency —Parametersthat measure the center of a frequencydistribution.

Measures of variation —Parameters thatmeasure the tendency of scores to be close orfar away from the center point.

Median —The center of a distribution ofscores, such that half of the scores are abovethat number, and half of the scores in the dis-tribution are below that number.

Mode —The most frequently occurring score.

Outliers —These are deviant scores that havebeen legitimately gathered and are not due toequipment failures, yet they are uncommonscores or they range far from the central ten-dency of the data set.

Range —The lowest score and the highestscore in a distribution.

Sampling distribution —A theoretical fre-quency distribution based on repeated sam-pling of n-sized samples from a givenpopulation. If the means of these samples areformed into a frequency distribution, theresult is a sampling distribution.

Standard deviation (σ or SD) —A parame-ter of variability of data about the mean score.

Standard error of the mean —The standard

deviation of a sampling distribution of means.

Unbiased estimator —A sample parameter

that neither overestimates nor underesti-mates the population value.

Uninterpretable results —A condition when a test’s outcome is invalid because it hasnot been given according to its correct instruc-tions or directions.

Variance (σ2) —A parameter of variability ofdata about the mean score, which is thesquare of the standard deviation.

Chapter 3 Practice Problems

1. Find the mean, median, and mode for each of the following sets of scores:

a. 35, 55, 80, 72, 55, 66, 74

b. 110, 115, 102, 102, 107, 102, 108, 110




c. 21, 19, 18, 30, 16, 30

d. 24, 26, 27, 22, 23, 22, 26

2. To familiarize yourself with the measures of variation, compute the range, standard devia-

tion, and variance for each set of scores. a. 0.6, 0.5, 0.8, 0.6, 0.2, 0.3, 0.9, 0.9, 0.7

b. 9.62, 9.31, 9.15, 10.11, 9.84, 10.78, 9.08, 10.17, 11.23, 12.45

c. 1001, 1253, 1234, 1171, 1125, 1099, 1040, 999, 1090, 1066, 1201, 1356

3. State the differences between klinkers and outliers. How do statisticians deal with each?

4. State the three predictions made by the empirical rule.

SPSS Lesson 3

Your objective for this assignment is to become familiar with generating measurements of centraltendency and variation using SPSS.

Generating Central Tendency and Variation Statistics

Follow these steps to open the program and create a frequency distribution graph for the vari-

able Age in the Schizoid-Aspergers-Controls SPSS.sav data file, available online at www.uccs

.edu/~faculty/fcoolidg//Schizoid-Aspergers-Controls SPSS.sav.

1. In the Data Editor , click File > Open and choose the Schizoid-Aspergers-Controls SPSS.

sav data set that you downloaded to your desktop.




2. Click Analyze > Descriptive Statistics > Frequencies to open the Frequencies dialog.

3. Double-click the Current Age [Age] variable to move it to the right into the (selected)

Variable(s) field.

4. Click Statistics to open the Frequencies: Statistics dialog.

5. In the Central Tendency group box, select Mean, Median, and Mode.

6. In the Dispersion group box, select Std. deviation, Variance, Range, Minimum, and

Maximum.




7. Click Continue > OK .

8. This opens the Statistics Viewer to display the frequency distribution for the Age variable.

9. Observe the statistics table for the Current Age variable in the Statistics Viewer .




Note that the minimum value is 3 and the maximum value is 16, consistent with data col-

lected from participants between the ages of 3 and 16. Also note that the variable contains 71

valid entries with 0 missing entries. The data range is 13, as would be expected between the ages

of 3 and 16.

The mean (average) age is 8.75 years, the median (half of the data above this value, halfof the data below this value) age is 8.00. If this was a perfectly normal distribution, the mean

and the median would be the same. Because the mean is slightly greater than the median, you

know that the data are positively skewed. The mode score is the value with the highest fre-

quency. As you can see in the Current Age table below, the value 4 has the highest frequency

(9 instances), therefore, the mode of the data is 4. The sample standard deviation is 4.204.

This means that about 66% of the data lie between the mean value (8.75) plus or minus the

standard deviation (4.204). For this data set, about 66% of the children are between 4.546 and

12.954 years or are roughly 4.5 to 13 years old.

The variance is calculated by squaring the standard deviation: 4.2042 = 17.678.

10. Observe the frequency distribution table for Current Age variable.



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Statistics a Gentle Introduction Ch_3